You can just watch the network traffic from the device to see when it is streaming audio.
Now this obviously doesn't rule out the device saving stuff locally, but plenty of people have verified that it does generally only send lots of data to the server when you have just asked it to wake up.
Although that does not rule out the possibility that it stores data locally and only transmit the recorded payload (since the time of last transmission) with the next legitimate payload when the wake word is spoken.
Not saying that they do though since if they were discovered to be doing that, it's way too damaging for Amazon's reputation for it to be worth going through the trouble.
It is possible it's doing local voice recognition on all audio, and just sending the text in their payloads. If the content is encrypted, there's no way for any network snooping to determine what they're sending.
This is software... saying there's "no way" is usually incorrect.
For example, you could root the device and disable the encryption.
Or, you could perform experiments in a controlled setting where you play a series of identical recordings to Alexa and measure the statistical similarity in the outgoing data for each. The encryption scheme probably provides more entropy than just the audio, so maybe statistical analysis wouldn't help, but it's a start.
Now this obviously doesn't rule out the device saving stuff locally, but plenty of people have verified that it does generally only send lots of data to the server when you have just asked it to wake up.